Armance Larfeuille, Katia Voltz, Manoël Pidoux, Elodie Shoeiline Kwan and Nina Bidet

Introduction

We are doing this report as part of our “Machine Learning” course in which we have to produce a detailed report based on an original database. To do this, we will re-analyze an existing data set by adding new elements (new models, features, outcome labels, approaches, interpretations, etc). Our goal is to classify mountains according to different features. To do this, we will go though supervised and unsupervised learning methods.

Overview and motivation

After several searches, we found a very interesting article explaining the interactions between the impacts of climate, soil and biotic on the interplay of the different facets of Alpine plant diversity. The scientists collected these data on three mountains in different locations and with opposite characteristics.

‘Guadarrama - Sierra de Guadarrama’ in Spain with a Mediterranean climate
‘Ordesa ~ Monte Perdido - Central Pyrenees’ in Spain with a Temperate climate
‘Central Andes’ in Chile with a Mediterranean climate

The database provided with the article was detailed, interesting and included enough observations to conduct our project. An analysis has already been carried out on this data set and we will complete it by using the skills acquired during our semester of course. Indeed, we will add machine learning methods of classification and regression.

Article: https://www.sciencedirect.com/science/article/pii/S2352340919311710
Data : 1-s2.0-S2352340919311710-mmc1.xlsx

Previous analysis explaination:

In the referenced article, different analysis have been made on the facets of alpine plant diversity; functional (FD), phylogenetic (PD) and taxonomic diversity (TD), in the three different mountains and with contrasting evolutionary histories and climate conditions. It means that the used database contains the plant species and the mountain characteristics. Their goal is predict the different diversity facet (response variables Y) thanks to the mountain features; Elevation, Potential solar radiation, Solar total nitrogen. The method used are Cluster analysis and SEM model.

Our project does not take into account the plants. THe goal is to study the mountains only based on the soil characteristics.

Project objectives

Our goal is to use the relevant machine learning tools, with supervised and unsupervised learning methods to characterize our data frame. In our case, it would be a question of being able to determine which mountain it is according to the given soil characteristics.

Structure of the report

Introduction
Part 1 : Data preparation
Part 2 : Exploratory Data Analysis (EDA)
Part 3 : Analysis
- Supervised learning analysis
- Unsupervised learning analysis
Conclusion

Part 1 : Data Preparation

In this first part, we will first download the data into R. Then, we will apply the necessary transformations on these data so that they can be correctly exploited for the following analysis.

Import dataset

Below is the first 4 observations of the datasets.

Sample_ID	Country	Muntain Range	Locality	Plot	Plot_ID	Subplot	Date	Day	Month	Year
1	Spain	Sierra de Guadarrama	Asomate-Hoyos	1	1	1	2010-07-21	21	7	2010
2	Spain	Sierra de Guadarrama	Asomate-Hoyos	1	1	2	2010-07-21	21	7	2010
3	Spain	Sierra de Guadarrama	Asomate-Hoyos	1	1	3	2010-07-21	21	7	2010
4	Spain	Sierra de Guadarrama	Asomate-Hoyos	1	1	4	2010-07-21	21	7	2010
5	Spain	Sierra de Guadarrama	Asomate-Hoyos	1	1	5	2010-07-21	21	7	2010
6	Spain	Sierra de Guadarrama	Bola del Mundo	2	2	1	2010-06-22	22	6	2010

Grid zone	UTM_X	UTM_Y	Elevation	Orientation	Slope	Radiation	Phos_P	Glu_P	SOC_P	NT_P
30 T	425862.7	4517724	2225	175	4	0.8088464	4.437515	2.505851	6.315554	4.070239
30 T	425862.7	4517724	2225	175	4	0.8088464	4.437515	2.505851	6.315554	4.070239
30 T	425862.7	4517724	2225	175	4	0.8088464	4.437515	2.505851	6.315554	4.070239
30 T	425862.7	4517724	2225	175	4	0.8088464	4.437515	2.505851	6.315554	4.070239
30 T	425862.7	4517724	2225	175	4	0.8088464	4.437515	2.505851	6.315554	4.070239
30 T	417601.1	4515745	2242	182	2	0.7879971	5.171000	3.234583	5.092630	4.546233

PT_P	K_P	pH_P	Cond_P	Phos_B	Glu_B	SOC_B	NT_B	PT_B	K_B
0.456501	0.0086492	4.925	31.284	2.888396	1.691185	3.762122	3.297689	0.4450694	0.0025165
0.456501	0.0086492	4.925	31.284	2.888396	1.691185	3.762122	3.297689	0.4450694	0.0025165
0.456501	0.0086492	4.925	31.284	2.888396	1.691185	3.762122	3.297689	0.4450694	0.0025165
0.456501	0.0086492	4.925	31.284	2.888396	1.691185	3.762122	3.297689	0.4450694	0.0025165
0.456501	0.0086492	4.925	31.284	2.888396	1.691185	3.762122	3.297689	0.4450694	0.0025165
3.923043	0.0133951	5.232	48.020	3.102327	1.955476	2.993161	3.168254	3.1682544	0.0067595

pH_B	Cond_B	Phos_T	Glu_T	SOC_T	NT_T	PT_T	K_T	pH_T	Cond_T
5.402	22.198	3.593446	2.203418	5.736881	3.958871	0.4871885	0.0059856	5.214310	27.60922
5.402	22.198	3.593446	2.203418	5.736881	3.958871	0.4871885	0.0059856	5.214310	27.60922
5.402	22.198	3.593446	2.203418	5.736881	3.958871	0.4871885	0.0059856	5.214310	27.60922
5.402	22.198	3.593446	2.203418	5.736881	3.958871	0.4871885	0.0059856	5.214310	27.60922
5.402	22.198	3.593446	2.203418	5.736881	3.958871	0.4871885	0.0059856	5.214310	27.60922
5.760	28.700	5.607705	3.481617	5.288592	4.912220	4.8853870	0.0153168	5.505655	48.93783

We are deleting the variables that are not useful for use. The variables such as the UTM grid zone, UTM_X and UTM_Y are related to the samples and we are not wanting to make predictions based on those characteristics but rather on the actual soil composition. The variable Elevation, slope, plot_ID, Orientation will not help us neither because it is related to the place where the sample has been taken. It is a certain amount above the some tree limits. We rename the variable Muntain Range as Mountain_range to facilitate future use.

After these changes, he database Mountain_data is composed of 430 observation and 34 variables.

Country : chr Spain, Chile
Mountain_range : chr Sierra de Guadarrama, Central Pyrenees
Locality : chr Locality of the sample (there is different location on the same mountain)
Plot : num In each site a sampling plot of 20 × 20 m was etablished in a relatively homogeneous vegetation area.
Subplot : num There is 5 subplots in each plot
Date : Date Date of the sample
Day : num Day of the sample
Month : num Month of the sample
Year : num Year of the sample
Radiation : num Potential solar radiation (0-1)
Phos_P : num Phosphatase enzyme. Measured on dry soil sampled under plant (µmol/gr dry soil/h)
Glu_P : num β-glucosidase enzyme. Measured on dry soil sampled under plant canopy (µmol/gr dry soil/h)
SOC_P : num soil organic carbon. Measured on dry soil sampled under plant canopy (%)
NT_P : num soil total nitrogen. Measured on dry soil sampled under plant canopy (mg/g soil)
PT_P : num available phosphorus. Measured on dry soil sampled under plant canopy (mg/g soil)
K_P : num potassium content. Measured on dry soil sampled under plant canopy (mg/g soil)
pH_P : num pH. Measured on dry soil sampled under plant canopy
Cond_P : num electrical conductivity. Measured on dry soil sampled under plant canopy (µS/cm)
Phos_B : num Phosphatase enzyme. Measured on dry soil sampled from bare soil areas (µmol/gr dry soil/h)
Glu_B : num β-glucosidase enzyme. Measured on dry soil sampled from bare soil areas (µmol/gr dry soil/h)
SOC_B : num soil organic carbon. Measured on dry soil sampled from bare soil areas (%)
NT_B : num soil total nitrogen. Measured on dry soil sampled from bare soil areas (mg/g soil)
PT_B : num available phosphorus. Measured on dry soil sampled from bare soil areas (mg/g soil)
K_B : num potassium content. Measured on dry soil sampled from bare soil areas (mg/g soil)
pH_B : num pH. Measured on dry soil sampled from bare soil areas
Cond_B : num electrical conductivity. Measured on dry soil sampled from bare soil areas (µS/cm)
Phos_T : num Phosphatase enzyme. Values averaged and weighted by the respective cover of bare ground and vegetated area in each site (µmol/gr dry soil/h)
Glu_T : num β-glucosidase enzyme. Values averaged and weighted by the respective cover of bare ground and vegetated area in each site (µmol/gr dry soil/h)
SOC_T : num soil organic carbon. Values averaged and weighted by the respective cover of bare ground and vegetated area in each site (%)
NT_T : num soil total nitrogen. Values averaged and weighted by the respective cover of bare ground and vegetated area in each site (mg/g soil)
PT_T : num available phosphorus. Values averaged and weighted by the respective cover of bare ground and vegetated area in each site (mg/g soil)
K_T : num potassium content. Values averaged and weighted by the respective cover of bare ground and vegetated area in each site (mg/g soil)
pH_T : num pH. Values averaged and weighted by the respective cover of bare ground and vegetated area in each site
Cond_T : num electrical conductivity. Values averaged and weighted by the respective cover of bare ground and vegetated area in each site (µS/cm)

Now that we have the variables we want to work with, we see that they are all considered as numeric. However, we know that some of them should be categorical as they have different levels. We are going to transform Country, Mountain_Range, Locality, Plot and Subplot as factors.

We can export this cleaned dataset on the data folder of our project.

write.csv(Mountain_data,"../data/Mountain_data_cleaned.csv", row.names = FALSE)

For a better representation of the data we created 3 maps with each icon representing a subsample belonging to a larger sample. Each map represents a mountain region, two in Spain and one in Chile.

Sierra de Guadarrama (Spain)

Mediterranean climate

Central Pyrenees (Spain)

temperate climate

Chilean Andes (Chile)

Mediterranean climate

Part 2 : Exploratory Data Analysis

This section is dedicated to understanding the data. We will provide an analysis of the data set using a visual approach in order to summarize their main characteristics.

We will analyze some basic statistical elements for each variable. To do this we need to transform the variable Date to date format.

Below, we look at the general aspect of the data set and try to discover if there are any missing values.

rows	columns	all_missing_columns	total_missing_values	complete_rows	total_observations
430	34	0	2	428	14620

There are 2 missing values in the feature Glu_P. One is at the instance number 377 and the other one is at the instance 378.

	Country	Mountain_range	Locality	Plot	Subplot	Date	Glu_P
377	Chile	Central Andes	Baños de Colinas	76	2	2014-01-21	NA
378	Chile	Central Andes	Baños de Colinas	76	3	2014-01-21	NA

To understand the distribution of our data in the data set we use the following graph:

There is a predominance of the continous columns compared to the discrete columns. Most of our variables will use both discrete and continours features. We can also notice that the share of missing observations represents only 0.014% of the total number of observations. Which at first sight makes it a good data set.

We take our search for anomalies further by exploring the characteristics of each variable.For the readability of the report, we show only a few variables.

## Country 
##        n  missing distinct 
##      430        0        2 
##                       
## Value      Chile Spain
## Frequency    100   330
## Proportion 0.233 0.767

## Mountain_range 
##        n  missing distinct 
##      430        0        3 
##                                                                          
## Value             Central Andes     Central Pyrenees Sierra de Guadarrama
## Frequency                   100                  135                  195
## Proportion                0.233                0.314                0.453

## Phos_P 
##        n  missing distinct     Info     Mean      Gmd      .05      .10 
##      430        0      268        1    3.477     2.26   0.6777   1.0753 
##      .25      .50      .75      .90      .95 
##   1.9318   3.1503   4.7146   6.2213   7.1411 
## 
## lowest : 0.01980997 0.16225689 0.25041658 0.31268048 0.32430518
## highest: 7.73917664 7.90167428 8.05414937 8.64896403 8.64973041

## Glu_P 
##        n  missing distinct     Info     Mean      Gmd      .05      .10 
##      428        2      272        1    2.102     1.24   0.2986   0.5062 
##      .25      .50      .75      .90      .95 
##   1.3679   2.0922   2.7710   3.2561   3.9541 
## 
## lowest : 0.1074305 0.1110000 0.1140850 0.1150734 0.1619782
## highest: 4.5538748 4.7529319 5.2224173 5.5441612 6.3505287

## NT_P 
##        n  missing distinct     Info     Mean      Gmd      .05      .10 
##      430        0      273        1    3.971    2.688   0.4501   0.7760 
##      .25      .50      .75      .90      .95 
##   2.1575   3.9155   5.6767   7.2597   8.0787 
## 
## lowest :  0.1156901  0.1610857  0.1833210  0.2201596  0.2365415
## highest:  8.4518587  8.9110000  9.0285000 10.8160000 18.0010000

We can observe that there is a big difference between the total number of observations and the number of distinct observations for the variables related to the chemical elements. We will try to understand where this difference comes from in the visual analysis part.

Data visualization : Plotting the data

We plot the numerical variables.

Many variables appear with a distribution that looks like the log-normal distribution.

The variables pH_B and pH_T are not normally distributed.
Most of the variables are right-tailed.

We will then use box-plots to detect outliers on numerical variable and compare the distributions to each mountain class.

We can see the real differences between the mountains.

Generally speaking, it seems that the mountain “Sierra de Guadarrama” has a higher value of Glu_P, Phos_P, SOC_P, Glu_B, Phos_B, SOC_B, Glu_T, Phos_T and SOC_T.
There are a lot of outliers in almost all the features.

We then plot the categorical variables:

We see that more observations come from Spain, which is normal since two out of three mountains are located in Spain. The localities where the samples were taken are almost all composed of a sample of 5 subsamples. Some localities, perhaps more interesting for the study, were sampled several times, but always by a multiple of 5 subsamples.

We have more observations about the mountain “Sierra de Guadarrama” (195) compared to “Central Andes” (100) and “Central Pyrenees” (135). As the differences between the number of observation is big enough for us to be careful on the results and consider to balance the data if it is needed.

We can comment on the number of different observations and the effect on accuracy. We can also focus more on sensibility and sensitivy if needed since there is twice more informations on Sierra de Guadarrama.

As described above with the summary output of the data, we see that we have more information on the mountain “Sierra de Guadarrama”. There is twice more information compared to the mountain “Central Andes”. Our final result might be affected on a bad way because the model will tend to produce a good accuracy (so having a tendency to predict “Sierra de Guadarrama” more often) but it will not be good enough to predict a new instance.

We will have to see if we will need to balance our data to get a better model.

We will also inspect the possible duplicate observations, indeed as previously found, some variables do not have the whole of their observations which are distinct.

We notice immediately the poverty of the data concerning the samples of Sierra de Guadarrama, this function leaves us with a data set of only 274 observations. We will therefore try a first time to implement our models by keeping the duplicates, knowing that identical values in the train set and the test set will influence the measured accuracy of the model. Then we will test again our models with the reduced data set to observe if there is a loss of accuracy.

For the rest of the EDA we will continue the analysis on the complete data set.

From the correlation plot it seems that some pattern can be observed. The variables concerning the Phosphatase enzyme seems to be positively correlated with the variable about Soil organic carbon.

With this plot, we see indeed that the families of Soil organic carbon and Phosphatase enzyme are significantly positively correlated. The correlation coefficient going from 0.739 (SOC_B - Phos_P) to 0.947 (SOC_T - SOC_P).

Principal Component Analysis (PCA) exploration:

This analysis helps us to understand the link between the explanatory variables.

The first step is to analyse the data in the covarianve matrix as we did before, and where we found the positive correlation between the Soil organic carbon and Phosphatase enzyme.
The second step is to group the data into Principal Components.
The third step is to produce a variable factor map to better understand the role of each factor.

Second step - Principal Component Result:

## Importance of components:
##                           PC1    PC2    PC3     PC4     PC5     PC6     PC7
## Standard deviation     3.3080 2.2090 1.6387 1.26851 1.02032 0.97230 0.82856
## Proportion of Variance 0.4377 0.1952 0.1074 0.06436 0.04164 0.03781 0.02746
## Cumulative Proportion  0.4377 0.6329 0.7403 0.80469 0.84633 0.88414 0.91161
##                           PC8     PC9    PC10   PC11    PC12    PC13    PC14
## Standard deviation     0.7106 0.63948 0.58236 0.5050 0.41430 0.37199 0.32284
## Proportion of Variance 0.0202 0.01636 0.01357 0.0102 0.00687 0.00554 0.00417
## Cumulative Proportion  0.9318 0.94816 0.96173 0.9719 0.97879 0.98433 0.98850
##                           PC15    PC16    PC17    PC18    PC19   PC20    PC21
## Standard deviation     0.25571 0.23393 0.21525 0.17576 0.17150 0.1415 0.12739
## Proportion of Variance 0.00262 0.00219 0.00185 0.00124 0.00118 0.0008 0.00065
## Cumulative Proportion  0.99111 0.99330 0.99515 0.99639 0.99757 0.9984 0.99902
##                          PC22    PC23    PC24    PC25
## Standard deviation     0.1115 0.08012 0.05935 0.04711
## Proportion of Variance 0.0005 0.00026 0.00014 0.00009
## Cumulative Proportion  0.9995 0.99977 0.99991 1.00000

We see that the first component explain 43.77%% of the overall variation. -The second component explain a further 19.52%.
With a reduction of 4 principal components, we obtain a cumulative variance of 80.5%, superior to the threshold of 75%.
The rest of the components (Components 4 to 25) explain 19.5% overall.

Here, as the command prcomp do not allow NAs in the data. We use the command na.omit on our reduced data containing the numerical values to omit all NAs cases from the data frame.

For the further analysis, we can study as well the eigenvalues in order to select a good number of components.

Eigenvalue analysis:

##          eigenvalue variance.percent cumulative.variance.percent
## Dim.1  10.936952114     43.747808455                    43.74781
## Dim.2   4.863450112     19.453800446                    63.20161
## Dim.3   2.694400932     10.777603728                    73.97921
## Dim.4   1.602728782      6.410915127                    80.39013
## Dim.5   1.045432762      4.181731049                    84.57186
## Dim.6   0.954822102      3.819288408                    88.39115
## Dim.7   0.688518236      2.754072944                    91.14522
## Dim.8   0.503178240      2.012712959                    93.15793
## Dim.9   0.411065663      1.644262652                    94.80220
## Dim.10  0.336691388      1.346765554                    96.14896
## Dim.11  0.254731126      1.018924506                    97.16789
## Dim.12  0.172594714      0.690378855                    97.85826
## Dim.13  0.138243331      0.552973325                    98.41124
## Dim.14  0.105400519      0.421602078                    98.83284
## Dim.15  0.067346066      0.269384264                    99.10222
## Dim.16  0.054954619      0.219818475                    99.32204
## Dim.17  0.046558386      0.186233542                    99.50828
## Dim.18  0.031588992      0.126355968                    99.63463
## Dim.19  0.029601829      0.118407318                    99.75304
## Dim.20  0.020191077      0.080764308                    99.83380
## Dim.21  0.016267999      0.065071996                    99.89888
## Dim.22  0.012623924      0.050495696                    99.94937
## Dim.23  0.006886114      0.027544454                    99.97692
## Dim.24  0.003535893      0.014143571                    99.99106
## Dim.25  0.002235080      0.008940321                   100.00000

We obtain the cumulative variance, as before, and also the eigenvalues.

A first rule of thumb is to stop adding components whent the total variance explained exceeds a high value, like 80% for example.
Another rule is the Kaiser-Guttman rule which states that components with an eigenvalue greater than 1 should be retained. The reason for this is we have ‘p’ variables so the sum of the eigenvalues is ‘p’. A value above 1 is above average.

Therefore, we can consider the dimension from 1 to 5:

Cumulative variance: 84.57%
Eigenvalue > 1

Screeplot of eigenvalue.

A screeplot represents the values of each eigenvalue.
According to the Kaiser-Guttman rule, we shold stop at Component 5.

Third step - Variable Factor Map:

The variable factor map show the variables and organized them along dimensions. Here the first two dimensions are represented.

Other representation:

Dimension 1 (x-axis): highly correlated to Phos_T, Phos_B, Phos_P and Glu_T
Dimension 1 is moderately correlated to PT_B
Dimension 1 is poorly correlated to Cond_T and Cond_B.
Dimension 2 is well correlated to Cond_T and Cond_B.
Dimension 2 is also moderatly negatively correlated to Radiation.
It seems that we have 4 groups of variables playing a different role. On these two dimensions we notice that the mountain classes already separate into 3 distinct clusters

Indeed, Sierra de Guadarrama is more positively correlated to the Dim1
Central Andes is porely negatively correlated to the Dim 1, bur more negatively correlated to Dim 2: Negatively with Cond_T
Central Pyrenees is negatively correlated to Dim 1, and positively correlated to Dim 2.

Study of each variable according to the 5 dimensions:

Dim 1: Highly correlated to the PHOS, SOCand GLU
Dim 2: Correlated with Cond_P and Cond_T
Dim 3: Correlated with PT_P, PT_B and PT_T
Dim 4: Moderately correlated to K_B and K_T
Dim 5: Correlated to Radiation

Put into representation the 5 dimensions:

The square cosine shows the importance of a component for a given observation. It is therefore normal that observations close to the origin are less significant than those far from it. Here we decided to represent only one variable of each type since the same chemical elements tend to have the same behavior independently of their sampling method. A variable that has an interesting behavior is Radiation, indeed the more we select high dimensions the more this variable becomes important (except for dimension 4), while the variables related to chemical elements tend to decrease. Thus, we find radiation strongly correlated with dimension 5.

Analysis in 3D

As seen in the EDA, we can consider 5 dimensions. In the following graph we reduce in 3 dimensions the 3 mountains. Clusters may be apparent.

Through this 3D, we can observe the distribution of the 3 mountains in the PCA.Further in the analysis we will do a cluster analysis, to better understand the apparent classification between the mountains.

Part 3 : Analysis

Replace the NAs by the mean of the closest observations

Before starting with the analysis we know that some of the models we use do not work with NAs. To deal with them, we decided to replace the 2 NAs we have by the mean of their closest observations.

Splitting the data into Traning set and Test set

The next step is to split our data set into a training set (Mountain_data.tr_notsubs) and a test set (Mountain_data.te). The training set has 75% of the observations and the test set has the 25% remainder.

Balancing the Training set

As discussed in the EDA part, we should balance our data because we do not have the same amount of information on each mountains. We have more observations on Sierra de Guadarrama and half less on Central Andes.

We balanced the data on the training set Mountain_data.tr_notsubs.

## 
##        Central Andes     Central Pyrenees Sierra de Guadarrama 
##                   79                   79                   79

Now we see that all three mountains has the same amount of observations. We decided to sub-samples the observations because we think that duplicating them will create more biases as we do not have that many observations.

Our new training set is Mountain_data.tr.

Neural Network Model

We start by fitting a Neural Network model on our balanced training set (Mountain_data.tr). In order to choose the parameters of this neural network, we applied a simple hyperparameter tuning.

The best Neural Networks parameters would be to choose 4 hidden layers, with a decay of 0.1 since it is the best combination that gives the highest accuracy value.

The manually written Neural Network model

## # weights:  747
## initial  value 343.739521 
## iter  10 value 260.416957
## iter  20 value 260.320004
## iter  30 value 249.074172
## iter  40 value 200.294799
## iter  50 value 167.751554
## iter  60 value 147.603498
## iter  70 value 118.282740
## iter  80 value 115.248739
## iter  90 value 16.684235
## iter 100 value 15.105342
## final  value 15.105342 
## stopped after 100 iterations

Accuracy of Neural Network

## Confusion Matrix and Statistics
## 
##                       Reference
## Prediction             Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   21                0                    0
##   Central Pyrenees                 0               29                    0
##   Sierra de Guadarrama             0                0                   58
## 
## Overall Statistics
##                                      
##                Accuracy : 1          
##                  95% CI : (0.9664, 1)
##     No Information Rate : 0.537      
##     P-Value [Acc > NIR] : < 2.2e-16  
##                                      
##                   Kappa : 1          
##                                      
##  Mcnemar's Test P-Value : NA         
## 
## Statistics by Class:
## 
##                      Class: Central Andes Class: Central Pyrenees
## Sensitivity                        1.0000                  1.0000
## Specificity                        1.0000                  1.0000
## Pos Pred Value                     1.0000                  1.0000
## Neg Pred Value                     1.0000                  1.0000
## Prevalence                         0.1944                  0.2685
## Detection Rate                     0.1944                  0.2685
## Detection Prevalence               0.1944                  0.2685
## Balanced Accuracy                  1.0000                  1.0000
##                      Class: Sierra de Guadarrama
## Sensitivity                                1.000
## Specificity                                1.000
## Pos Pred Value                             1.000
## Neg Pred Value                             1.000
## Prevalence                                 0.537
## Detection Rate                             0.537
## Detection Prevalence                       0.537
## Balanced Accuracy                          1.000

With this confusion matrix command (confusionMatrix), we have more information on the model. As said before, we see that the accuracy is very high (99%) and we also see that we have a balanced accuracy of 1 which is the maximum we can get and which mean that our model do not suffer from unbalanced data.

Naive Bayes

The analysis of the “Naive Bayes” model comes back to the analysis of the density graphs of each variable by mountain range. From these density graphs we agree on what has been seen before, that the pH is a very good feature to classify the mountains. Indeed, pH_T ph_B and ph_P is medium in Central Andes, lower in Sierra de Guadarrama and higher in Central Pyrenees.

In addition to that, the observation of other variables such as phosphatase enzyme, (Phos_P, Phos_B, Phos_T), β-glucosidase enzyme (Glu_B, Glu_P, Glu_T) , soil organic carbon (SOC_T, SOC_B, SOC_P), soil total nitrogen (NT_P,NT_B,NT_T), electrical conductivity (Cond_P,Cond_B, Cond_T), and the radiation could allow us to get an idea of the mountain we are on. Indeed, a radiation lower than 0,6 indicates rather the Central Pyrenees. The electrical conductivity allows us to distinguish the Central Andes from the Central Pyrenees, since it is higher for the latter. If we look at the soil total nitrogen, we see that it is lower for the Central Andes than for the two others. The soil organic carbon allows us to distinguish the Central Andes from the Sierra de Guadarrama, since it is higher for this last one. The β-glucosidase enzyme is present at approximately the same level in Central Andes and Central Pyrenees but has a higher value in Sierra de Guadarrama. It is about the same for the phosphatase enzyme

However, some variables do not allow us to determine the mountain at all. It is the case by observing the phosphorus (PT_P, PT_B, PT_T) or the potassium content (K_P, K_B, K_T).

Accuracy of Naive bayes

## Confusion Matrix and Statistics
## 
##                       Reference
## Prediction             Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   17                0                    0
##   Central Pyrenees                 3               29                    0
##   Sierra de Guadarrama             1                0                   58
## 
## Overall Statistics
##                                           
##                Accuracy : 0.963           
##                  95% CI : (0.9079, 0.9898)
##     No Information Rate : 0.537           
##     P-Value [Acc > NIR] : < 2.2e-16       
##                                           
##                   Kappa : 0.9379          
##                                           
##  Mcnemar's Test P-Value : NA              
## 
## Statistics by Class:
## 
##                      Class: Central Andes Class: Central Pyrenees
## Sensitivity                        0.8095                  1.0000
## Specificity                        1.0000                  0.9620
## Pos Pred Value                     1.0000                  0.9063
## Neg Pred Value                     0.9560                  1.0000
## Prevalence                         0.1944                  0.2685
## Detection Rate                     0.1574                  0.2685
## Detection Prevalence               0.1574                  0.2963
## Balanced Accuracy                  0.9048                  0.9810
##                      Class: Sierra de Guadarrama
## Sensitivity                               1.0000
## Specificity                               0.9800
## Pos Pred Value                            0.9831
## Neg Pred Value                            1.0000
## Prevalence                                0.5370
## Detection Rate                            0.5370
## Detection Prevalence                      0.5463
## Balanced Accuracy                         0.9900

Through this confusion matrix, we can see that the Naive Bayes accuracy is of 96.3% and the balancced accuracy of 90.48%. Indeed it lower than the Neural Network, but it remain good. 3 Central Pyrenees have been mispredicted, and 1 Sierra de Guadarrama.

K-NN Model

We use a 2-NN to predict the test set using the training set

## Confusion Matrix and Statistics
## 
##                       Reference
## Prediction             Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   21                0                    0
##   Central Pyrenees                 0               29                    0
##   Sierra de Guadarrama             0                0                   58
## 
## Overall Statistics
##                                      
##                Accuracy : 1          
##                  95% CI : (0.9664, 1)
##     No Information Rate : 0.537      
##     P-Value [Acc > NIR] : < 2.2e-16  
##                                      
##                   Kappa : 1          
##                                      
##  Mcnemar's Test P-Value : NA         
## 
## Statistics by Class:
## 
##                      Class: Central Andes Class: Central Pyrenees
## Sensitivity                        1.0000                  1.0000
## Specificity                        1.0000                  1.0000
## Pos Pred Value                     1.0000                  1.0000
## Neg Pred Value                     1.0000                  1.0000
## Prevalence                         0.1944                  0.2685
## Detection Rate                     0.1944                  0.2685
## Detection Prevalence               0.1944                  0.2685
## Balanced Accuracy                  1.0000                  1.0000
##                      Class: Sierra de Guadarrama
## Sensitivity                                1.000
## Specificity                                1.000
## Pos Pred Value                             1.000
## Neg Pred Value                             1.000
## Prevalence                                 0.537
## Detection Rate                             0.537
## Detection Prevalence                       0.537
## Balanced Accuracy                          1.000

With this KNN supervised method, the prediction made on the test set gives us an accuracy of 1. It looks indeed perfect and attractive, but it is complytely biased by the fact that subplot can be identical to plot. Therefore, there is an overlay in the pbservations and the distance computed is 0. What we need to do is to make a unite dataset with no observations overlayed.

Analysis with unique dataset

We now want to replicate the above models with a another training set and test set that has been through a boostraping method. To do so, we first want to delete all the repeated observations from the original cleaned dataset.

Unique

We remove every replicated data with the “unique” function. We decided to do so to see if we will have better results. As we see above, the accuracy of our models are very high and this is normally unlikely. So we are testing if the high accuracy is due to the fact that we have duplicated observations in our dataset. (The scientists did sample different observations, but it comes out that the samples are exactly the same, they are maybe different at a very very small digits.)

Count

We have seen that the new Unique data set is largely unbalanced. The mountain that has the most information is now the one with the less observations, so information.

Sierra de Gaudarrama, after removing every replicated data with the “unique” function, only have 39 observations. It is unbalanced indeed, and insufficient to make accurate classification methods.

Split in a new training set and test set

Now that we have a new dataset, we have to split it again into a training set (Mountain_df.tr_notsubs) and a test set (Mountain_df.te ).

Random Forest

Once the duplicates are removed, we are left with a data set that is low in data, especially in the observations for the Sierra de Guadarrama class. With the poverty of our data we want to test an implementation of a random forest model. Indeed, the bagging should allow us to overcome the lack of data and give the best possible model. Another advantage of the random forest model is that it allows us to easily identify the importance of variables.

Here we can see the difference between the descending average of the precision that each variable adds to the model and the descending average of Gini which is a coefficient that shows the contribution of each variable to the homogeneity of nodes and leaves. We notice that some variables have a higher precision importance than their importance based on their Gini score, for our selection of future candidate variables for a simpler model we will make a compromise between these two measures.

From the variable importance we see that the variable pH_P, pH_T and pH_B are very important. We can then say that the pH in general is important.

##                       predtr
##                        Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   69                0                    0
##   Central Pyrenees                 0              107                    0
##   Sierra de Guadarrama             0                0                   29

## Accuracy of the train set = 1

The first table is the confusion matrix of the Random Forest model on the training set. Therefore it is normal to have an accuracy of 1 since the model already knows the observations.

##                       predte
##                        Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   31                0                    0
##   Central Pyrenees                 0               28                    0
##   Sierra de Guadarrama             0                0                   10

## Accuracy of the test set = 1

Then the second table is the confusion matrix of the random forest model on the test set. We still have a precision of 1, which makes it a good model. We have to keep in mind that the data set is very small and not necessarily representative of a larger sample.

Currently the model has many variables and we notice that many of them are not important for mountain classification. We are therefore thinking of testing a model with the 5 main variables to avoid overloading the model. Having less variables would allow us to predict a larger set of data with less computational power and resources. Moreover, the time saving could also be done during sampling, since less chemical elements would be needed to predict the class.

##                       predtr
##                        Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   69                0                    0
##   Central Pyrenees                 0              107                    0
##   Sierra de Guadarrama             0                0                   29

## Accuracy of the train set = 1

##                       predte
##                        Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   31                0                    0
##   Central Pyrenees                 1               27                    0
##   Sierra de Guadarrama             0                0                   10

## Accuracy of the test set = 0.9855072

Even with this model composed of fewer variables we still see a good accuracy of the test set compared to the train set. Here the error comes from the fact that we will have more tendency to predict Central Andes since we cannot balance the data between the classes otherwise we will reduce too much the size of the data set.

Overall the random forest model seems to be a good model that can be used with fewer variables while keeping a high accuracy.

But we now bootstrap to re-test some of our models and check if the accuracy changes once the duplicate data is removed.

Bootstrap

We can now proceed to the bootstraping with 100 replicates

We can now replicate the models.

Neural Network Model (with Unique bootstraped data)

Simple hyperparameter tuning, this code takes time to run.

The best neural network should have 4 hidden units and a decay of 0.1. It is the same as before.

## # weights:  375
## initial  value 284.692893 
## iter  10 value 116.079753
## iter  20 value 57.538311
## iter  30 value 29.136661
## iter  40 value 22.652969
## iter  50 value 21.197795
## iter  60 value 20.339347
## iter  70 value 18.971778
## iter  80 value 18.386184
## iter  90 value 18.351844
## iter 100 value 18.341811
## final  value 18.341811 
## stopped after 100 iterations

Now that we fitted the model and made the prediction, we can use the confusion matrix to see is the model performed well or not.

## Confusion Matrix and Statistics
## 
##                       Reference
## Prediction             Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   24                0                    0
##   Central Pyrenees                 0               39                    0
##   Sierra de Guadarrama             0                0                   10
## 
## Overall Statistics
##                                      
##                Accuracy : 1          
##                  95% CI : (0.9507, 1)
##     No Information Rate : 0.5342     
##     P-Value [Acc > NIR] : < 2.2e-16  
##                                      
##                   Kappa : 1          
##                                      
##  Mcnemar's Test P-Value : NA         
## 
## Statistics by Class:
## 
##                      Class: Central Andes Class: Central Pyrenees
## Sensitivity                        1.0000                  1.0000
## Specificity                        1.0000                  1.0000
## Pos Pred Value                     1.0000                  1.0000
## Neg Pred Value                     1.0000                  1.0000
## Prevalence                         0.3288                  0.5342
## Detection Rate                     0.3288                  0.5342
## Detection Prevalence               0.3288                  0.5342
## Balanced Accuracy                  1.0000                  1.0000
##                      Class: Sierra de Guadarrama
## Sensitivity                                1.000
## Specificity                                1.000
## Pos Pred Value                             1.000
## Neg Pred Value                             1.000
## Prevalence                                 0.137
## Detection Rate                             0.137
## Detection Prevalence                       0.137
## Balanced Accuracy                          1.000

KNN model (with Unique bootstraped data)

## Confusion Matrix and Statistics
## 
##                       Reference
## Prediction             Central Andes Central Pyrenees Sierra de Guadarrama
##   Central Andes                   24                0                    0
##   Central Pyrenees                 0               39                    3
##   Sierra de Guadarrama             0                0                    7
## 
## Overall Statistics
##                                           
##                Accuracy : 0.9589          
##                  95% CI : (0.8846, 0.9914)
##     No Information Rate : 0.5342          
##     P-Value [Acc > NIR] : 5.772e-16       
##                                           
##                   Kappa : 0.9281          
##                                           
##  Mcnemar's Test P-Value : NA              
## 
## Statistics by Class:
## 
##                      Class: Central Andes Class: Central Pyrenees
## Sensitivity                        1.0000                  1.0000
## Specificity                        1.0000                  0.9118
## Pos Pred Value                     1.0000                  0.9286
## Neg Pred Value                     1.0000                  1.0000
## Prevalence                         0.3288                  0.5342
## Detection Rate                     0.3288                  0.5342
## Detection Prevalence               0.3288                  0.5753
## Balanced Accuracy                  1.0000                  0.9559
##                      Class: Sierra de Guadarrama
## Sensitivity                              0.70000
## Specificity                              1.00000
## Pos Pred Value                           1.00000
## Neg Pred Value                           0.95455
## Prevalence                               0.13699
## Detection Rate                           0.09589
## Detection Prevalence                     0.09589
## Balanced Accuracy                        0.85000

Now, the accuracy is of 95.89%, lower than before, but might be more realistic with the unique method and the bootstrap. The balanced accuracy remains equal to 1.

Unsupervised learning method

Cluster Analysis

Agglomerative Hierarchical Clustering (AGNES) with Manhattan distance

Using the dendrogram with complete linkage on Manhattan distance with the silhouette method, we identified k=5 as the optimal number of clusters.

By analyzing this clusters, we can differentiate some of them. Firstly, cluster 1 seems to be more or less in the average of the other clusters for all variables, except for its pH, (pH_B, pH_P and pH_T) which is well below the others. Cluster 2 is distinguished by its high PT_P, PT_T and PT_B. Cluster 3 is characterized by a high pH_B, pH_P and pH_T and a low β-glucosidase enzyme (Glu_B, Glu_P and Glu_T) content (like cluster 5). By looking at the distribution of cluster 4 we see that it contains few observations but that it is well distinguished from the others. Indeed, only one line appears where the median merges with the width of its distribution. This cluster has the lowest radiation, the highest soil organic carbon (SOC_B, SOC_T and SOC_P) and highest soil total nitrogen (NT_T, NT_B and NT_P). Finally, the last cluster has a pH (pH_B, pH_P and pH_T) close to zero and a high radiation. It has oservations with the lowest soil organic carbon (SOC_B, SOC_T and SOC_P) On the silhouette plot, we see that the average silhouette considering the all five clusters is 0.45. Cluster 1 and 4 are the most homogeneous clusters and with the highest silhouette, respectively, 0,56 and 0,70. Cluster 5 has some mountains badly clustered as it has negative silhouettes, and the lowest silhouette (0,15).

Final analysis of score:

Compare the model scores to select the most accurate one

To recap, we studied the following models with a balanced dataset:

Neural network, with an accuracy and balanced accuracy of 1
Naive Bayes, with an accuracy of 96.3% and balanced of 90,48%
Key Nearest Neighbor (K=2), with an accuracy and balanced accuracy of 1. We explained why this accuracy might be affect by the overlayed observations.

Then we fitted the models on an unique dataset, to avoid the overlayed observations.

Random Forest: The accuracy appeared to be equal to 1 as well
Neural Network with unoque bootstraped dataset: Accuracy of 1
KNN (K=2): Accuracy of 95.89% and Balanced accuracy of 1

To sum up, all the models have an accuracy above 90%. Neural Network seems to perfom very well, before and after the unique data set. We have to bear in mind the size of the data set, the overlayed observations and the overfitting problem.

Choose a model and say why

For modeling part : Concerning the ROC metric and more precisely the AUC : ###

Limitations of our model:

The main limitations come from our data set used to train the model and test it. Indeed, as we noticed during the EDA we have a lot of duplicate data for the Sierra de Guadarrama region, these data when not removed from the data set influence the accuracy of both the test set and the train set. It is obvious that the prediction of identical data leads to a good result. Moreover, even if we remove these duplicates, the fact that the sub samples were taken close to each other prevents us from having results that are representative of a real study that would be conducted with samples taken from more widely spaced areas. We therefore have a false good precision, since the values to predict are close to those used to train the model. Ideally we should use only one sub sample per sample for our model, but in this case we would be sorely lacking in data to train our model.

Future work:

in the analysis, we discovered a strong positive correlation between Soil organic carbon and Phosphatase enzyme. For the future analysis, it would be interesting to see the features and characteristics that influence these two variables.

We found an interesting articles describing the effects of phosphate rock and organic inputs on soil organic carbon and acid phosphatase activity.

Title: Soil Organic Carbon and Acid Phosphatase Enzyme Activity Response to Phosphate Rock and Organic Inputs in Acidic Soils of Central Highlands of Kenya in Maize: https://ir-library.ku.ac.ke/bitstream/handle/123456789/22271/Soil%20Organic%20Carbon%20and%20Acid%20Phosphatase%20Enzyme....pdf?sequence=1

With more time an interesting study could be to look on the map if the samples were taken close to a watercourse by transforming this information into a binary variable. Then we can analyze if there is a correlation between the chemical elements of the soil and their proximity to a water point, independently of the region and the climate. Thus, depending on the result of this analysis, we will be able to specify the description of the quality of a sample in terms of its ability to predict a class. The operation could be repeated with the samples close to a forest, as well as with the altitude of sampling.

Machine Learning

Soil analysis on 3 different mountain ranges

23 mai, 2022

Introduction

Overview and motivation

Previous analysis explaination:

Project objectives

Structure of the report

Related work

Part 1 : Data Preparation

Import dataset

Sierra de Guadarrama (Spain)

Mediterranean climate

Central Pyrenees (Spain)

temperate climate

Chilean Andes (Chile)

Mediterranean climate

Part 2 : Exploratory Data Analysis

Data visualization : Plotting the data

Principal Component Analysis (PCA) exploration:

Second step - Principal Component Result:

Eigenvalue analysis:

Screeplot of eigenvalue.

Third step - Variable Factor Map:

Other representation:

Study of each variable according to the 5 dimensions:

Put into representation the 5 dimensions:

Analysis in 3D

Part 3 : Analysis

Replace the NAs by the mean of the closest observations

Splitting the data into Traning set and Test set

Balancing the Training set

Neural Network Model

Accuracy of Neural Network

Naive Bayes

Accuracy of Naive bayes

K-NN Model

Analysis with unique dataset

Unique

Count

Split in a new training set and test set

Random Forest

Bootstrap

Neural Network Model (with Unique bootstraped data)

KNN model (with Unique bootstraped data)

Unsupervised learning method

Cluster Analysis

Agglomerative Hierarchical Clustering (AGNES) with Manhattan distance

Final analysis of score:

Compare the model scores to select the most accurate one

Limitations of our model:

Future work: